Phylogenetic degrees for claw trees. (English) Zbl 07852601
Summary: Group-based models appear in algebraic statistics as mathematical models coming from evolutionary biology, namely in the study of mutations of genomes. Motivated also by applications, we are interested in determining the algebraic degrees of the phylogenetic varieties coming from these models. These algebraic degrees are called phylogenetic degrees. In this paper, we compute the phylogenetic degree of the variety \(X_{G,n}\) with \(G\in \{ \mathbb{Z}_2, \mathbb{Z}_2 \times \mathbb{Z}_2, \mathbb{Z}_3\}\) and any \(n\)-claw tree. As these varieties are toric, computing their phylogenetic degree relies on computing the volume of their associated polytopes \(P_{G,n}\). We apply combinatorial methods and we give concrete formulas for these volumes.
MSC:
| 13F65 | Commutative rings defined by binomial ideals, toric rings, etc. |
| 05E40 | Combinatorial aspects of commutative algebra |
| 14M25 | Toric varieties, Newton polyhedra, Okounkov bodies |
| 92Dxx | Genetics and population dynamics |
| 52Bxx | Polytopes and polyhedra |
| 13P25 | Applications of commutative algebra (e.g., to statistics, control theory, optimization, etc.) |
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